On the role of three functions in the Riemann literatures

Abstract

In Riemann's paper and Nachlass on the zeta function, $\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ has three different functional expressions, of which $$\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)=2\Re[\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})f(s)],\qquad\Re(s)=1/2$$ still has no literature to study it so far. Based on its geometric meaning, we obtain the number of zeros of the Riemann zeta function on the critical line is $$\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+\frac{\arg{f(1/2+iT)}}{\pi}+O(T^{-1}).$$ Research shows that Riemann's assertion about ~"One now finds indeed approximate this number of real roots within these limits" comes from this functional expression of $\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ which associated with the Jacobi function. Finally, this paper analyzes the reason why these conclusions are neglected.Comment: AMS-LaTeX v2.2, 11 page